Why the difference in rewards is so close between first and last places?
Because the worst reward of a given league must be better than the best reward of the league below. The main reason to try to win a given Battleground is to get to the next league, as the league alone has a pretty big impact on your rewards, not only placement rewards, but also the forge point drops you occasionally get. The higher the league, the more FP and Diamonds you can get (diamonds are 5 in Copper league, and +5 for each league higher). The chance to get 5 units is low in Copper league, but this chance is increased up until Diamond League.
The higher leagues have the best rewards. To move in the leagues, you need to gain MMR, which means that if you finish a battleground in a very good position, you will gain more MMR, thus you’ll move faster to the next leagues.